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1.

図書
図書
1. Riemannian geometry
Luther Pfahler Eisenhart
出版情報: Princeton : Princeton Univ. Press, [n. d.].  vii, 306 p. ; 24 cm
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2.

図書
図書
2. An introduction to Riemann-Finsler geometry (: hbk)
D. Bao, S.-S. Chern, Z. Shen
出版情報: New York : Springer-Verlag, c2000  xx, 431 p. ; 25 cm
シリーズ名: Graduate texts in mathematics ; 200
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目次情報: 続きを見る
Preface
Acknowledgments
Finsler Manifolds and Their Curvature / Part 1：
Finsler Manifolds and the Fundamentals of Minkowski Norms / Chapter 1：
Physical Motivations / 1.0：
Finsler Structures: Definitions and Conventions / 1.1：
Two Basic Properties of Minkowski Norms / 1.2：
Euler's Theorem / 1.2 A.：
A Fundamental Inequality / 1.2 B.：
Interpretations of the Fundamental Inequality / 1.2 C.：
Explicit Examples of Finsler Manifolds / 1.3：
Minkowski and Locally Minkowski Spaces / 1.3 A.：
Riemannian Manifolds / 1.3 B.：
Randers Spaces / 1.3 C.：
Berwald Spaces / 1.3 D.：
Finsler Spaces of Constant Flag Curvature / 1.3 E.：
The Fundamental Tensor and the Cartan Tensor / 1.4：
References for Chapter 1
The Chern Connection / Chapter 2：
Prologue / 2.0：
The Vector Bundle [pi]*TM and Related Objects / 2.1：
Coordinate Bases Versus Special Orthonormal Bases / 2.2：
The Nonlinear Connection on the Manifold TM \ 0 / 2.3：
The Chern Connection on [pi]*TM / 2.4：
Index Gymnastics / 2.5：
The Slash (...)[subscript / 2.5 A.：
Covariant Derivatives of the Fundamental Tensor g / 2.5 B.：
Covariant Derivatives of the Distinguished l / 2.5 C.：
References for Chapter 2
Curvature and Schur's Lemma / Chapter 3：
Conventions and the hh-, hv-, vv-curvatures / 3.1：
First Bianchi Identities from Torsion Freeness / 3.2：
Formulas for R and P in Natural Coordinates / 3.3：
First Bianchi Identities from "Almost" g-compatibility / 3.4：
Consequences from the dx[superscript k] [logical and] dx[superscript l] Terms / 3.4 A.：
Consequences from the dx[superscript k] [logical and] 1/F[delta]y[superscript 1] Terms / 3.4 B.：
Consequences from the 1/F[delta]y[superscript k] [logical and] 1/F[delta]y[superscript 1] Terms / 3.4 C.：
Second Bianchi Identities / 3.5：
Interchange Formulas or Ricci Identities / 3.6：
Lie Brackets among the [delta]/[delta]x and the F[characters not reproducible] / 3.7：
Derivatives of the Geodesic Spray Coefficients G[superscript i] / 3.8：
The Flag Curvature / 3.9：
Its Definition and Its Predecessor / 3.9 A.：
An Interesting Family of Examples of Numata Type / 3.9 B.：
Schur's Lemma / 3.10：
References for Chapter 3
Finsler Surfaces and a Generalized Gauss-Bonnet Theorem / Chapter 4：
Minkowski Planes and a Useful Basis / 4.0：
Rund's Differential Equation and Its Consequence / 4.1 A.：
A Criterion for Checking Strong Convexity / 4.1 B.：
The Equivalence Problem for Minkowski Planes / 4.2：
The Berwald Frame and Our Geometrical Setup on SM / 4.3：
The Chern Connection and the Invariants I, J, K / 4.4：
The Riemannian Arc Length of the Indicatrix / 4.5：
A Gauss-Bonnet Theorem for Landsberg Surfaces / 4.6：
References for Chapter 4
Calculus of Variations and Comparison Theorems / Part 2：
Variations of Arc Length, Jacobi Fields, the Effect of Curvature / Chapter 5：
The First Variation of Arc Length / 5.1：
The Second Variation of Arc Length / 5.2：
Geodesics and the Exponential Map / 5.3：
Jacobi Fields / 5.4：
How the Flag Curvature's Sign Influences Geodesic Rays / 5.5：
References for Chapter 5
The Gauss Lemma and the Hopf-Rinow Theorem / Chapter 6：
The Gauss Lemma / 6.1：
The Gauss Lemma Proper / 6.1 A.：
An Alternative Form of the Lemma / 6.1 B.：
Is the Exponential Map Ever a Local Isometry? / 6.1 C.：
Finsler Manifolds and Metric Spaces / 6.2：
A Useful Technical Lemma / 6.2 A.：
Forward Metric Balls and Metric Spheres / 6.2 B.：
The Manifold Topology Versus the Metric Topology / 6.2 C.：
Forward Cauchy Sequences, Forward Completeness / 6.2 D.：
Short Geodesics Are Minimizing / 6.3：
The Smoothness of Distance Functions / 6.4：
On Minkowski Spaces / 6.4 A.：
On Finsler Manifolds / 6.4 B.：
Long Minimizing Geodesics / 6.5：
The Hopf-Rinow Theorem / 6.6：
References for Chapter 6
The Index Form and the Bonnet-Myers Theorem / Chapter 7：
Conjugate Points / 7.1：
The Index Form / 7.2：
What Happens in the Absence of Conjugate Points? / 7.3：
Geodesics Are Shortest Among "Nearby" Curves / 7.3 A.：
A Basic Index Lemma / 7.3 B.：
What Happens If Conjugate Points Are Present? / 7.4：
The Cut Point Versus the First Conjugate Point / 7.5：
Ricci Curvatures / 7.6：
The Ricci Scalar Ric and the Ricci Tensor Ric[subscript ij] / 7.6 A.：
The Interplay between Ric and Ric[subscript ij] / 7.6 B.：
The Bonnet-Myers Theorem / 7.7：
References for Chapter 7
The Cut and Conjugate Loci, and Synge's Theorem / Chapter 8：
Definitions / 8.1：
The Cut Point and the First Conjugate Point / 8.2：
Some Consequences of the Inverse Function Theorem / 8.3：
The Manner in Which c[subscript y] and i[subscript y] Depend on y / 8.4：
Generic Properties of the Cut Locus Cut[subscript x] / 8.5：
Additional Properties of Cut[subscript x] When M Is Compact / 8.6：
Shortest Geodesics within Homotopy Classes / 8.7：
Synge's Theorem / 8.8：
References for Chapter 8
The Cartan-Hadamard Theorem and Rauch's First Theorem / Chapter 9：
Estimating the Growth of Jacobi Fields / 9.1：
When Do Local Diffeomorphisms Become Covering Maps? / 9.2：
Some Consequences of the Covering Homotopy Theorem / 9.3：
The Cartan-Hadamard Theorem / 9.4：
Prelude to Rauch's Theorem / 9.5：
Transplanting Vector Fields / 9.5 A.：
A Second Basic Property of the Index Form / 9.5 B.：
Flag Curvature Versus Conjugate Points / 9.5 C.：
Rauch's First Comparison Theorem / 9.6：
Jacobi Fields on Space Forms / 9.7：
Applications of Rauch's Theorem / 9.8：
References for Chapter 9
Special Finsler Spaces over the Reals / Part 3：
Berwald Spaces and Szabo's Theorem for Berwald Surfaces / Chapter 10：
Various Characterizations of Berwald Spaces / 10.0：
Examples of Berwald Spaces / 10.3：
A Fact about Flat Linear Connections / 10.4：
Characterizing Locally Minkowski Spaces by Curvature / 10.5：
Szabo's Rigidity Theorem for Berwald Surfaces / 10.6：
The Theorem and Its Proof / 10.6 A.：
Distinguishing between y-local and y-global / 10.6 B.：
References for Chapter 10
Randers Spaces and an Elegant Theorem / Chapter 11：
The Importance of Randers Spaces / 11.0：
Randers Spaces, Positivity, and Strong Convexity / 11.1：
A Matrix Result and Its Consequences / 11.2：
The Geodesic Spray Coefficients of a Randers Metric / 11.3：
The Nonlinear Connection for Randers Spaces / 11.4：
A Useful and Elegant Theorem / 11.5：
The Construction of y-global Berwald Spaces / 11.6：
The Algorithm / 11.6 A.：
An Explicit Example in Three Dimensions / 11.6 B.：
References for Chapter 11
Constant Flag Curvature Spaces and Akbar-Zadeh's Theorem / Chapter 12：
Characterizations of Constant Flag Curvature / 12.0：
Useful Interpretations of E and E / 12.2：
Growth Rates of Solutions of E + [lambda]E = 0 / 12.3：
Akbar-Zadeh's Rigidity Theorem / 12.4：
Formulas for Machine Computations of K / 12.5：
The Geodesic Spray Coefficients / 12.5 A.：
The Predecessor of the Flag Curvature / 12.5 B.：
Maple Codes for the Gaussian Curvature / 12.5 C.：
A Poincare Disc That Is Only Forward Complete / 12.6：
The Example and Its Yasuda-Shimada Pedigree / 12.6 A.：
The Finsler Function and Its Gaussian Curvature / 12.6 B.：
Geodesics; Forward and Backward Metric Discs / 12.6 C.：
Consistency with Akbar-Zadeh's Rigidity Theorem / 12.6 D.：
Non-Riemannian Projectively Flat S[superscript 2] with K = 1 / 12.7：
Bryant's 2-parameter Family of Finsler Structures / 12.7 A.：
A Specific Finsler Metric from That Family / 12.7 B.：
References for Chapter 12
Riemannian Manifolds and Two of Hopf's Theorems / Chapter 13：
The Levi-Civita (Christoffel) Connection / 13.1：
Curvature / 13.2：
Symmetries, Bianchi Identities, the Ricci Identity / 13.2 A.：
Sectional Curvature / 13.2 B.：
Ricci Curvature and Einstein Metrics / 13.2 C.：
Warped Products and Riemannian Space Forms / 13.3：
One Special Class of Warped Products / 13.3 A.：
Spheres and Spaces of Constant Curvature / 13.3 B.：
Standard Models of Riemannian Space Forms / 13.3 C.：
Hopf's Classification of Riemannian Space Forms / 13.4：
The Divergence Lemma and Hopf's Theorem / 13.5：
The Weitzenbock Formula and the Bochner Technique / 13.6：
References for Chapter 13
Minkowski Spaces, the Theorems of Deicke and Brickell / Chapter 14：
Generalities and Examples / 14.1：
The Riemannian Curvature of Each Minkowski Space / 14.2：
The Riemannian Laplacian in Spherical Coordinates / 14.3：
Deicke's Theorem / 14.4：
The Extrinsic Curvature of the Level Spheres of F / 14.5：
The Gauss Equations / 14.6：
The Blaschke-Santalo Inequality / 14.7：
The Legendre Transformation / 14.8：
A Mixed-Volume Inequality, and Brickell's Theorem / 14.9：
References for Chapter 14
Bibliography
Index
Preface
Acknowledgments
Finsler Manifolds and Their Curvature / Part 1：
3.

図書
図書
3. Riemannian geometry
by Luther Pfahler Eisenhart
出版情報: Princeton : Princeton Univ. Press, c1926  vii, 262 p. ; 24 cm
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4.

図書
図書
4. Geometry of Riemannian spaces
by Elie Cartan ; translated by James Glazebrook ; notes and appendices by R. Hermann
出版情報: Brookline, Mass. : Math Sci Press, c1983  xiv, 506 p. ; 24 cm
シリーズ名: Lie groups : history, frontiers, and applications ; [Series A] v. 13
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5.

図書
図書
5. Riemannian geometry during the second half of the twentieth century
Marcel Berger
出版情報: Providence, R.I. : American Mathematical Society, c2000  ix, 182 p. ; 26 cm
シリーズ名: University lecture series ; 17
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6.

図書
図書
6. Dirac operators in Riemannian geometry
Thomas Friedrich ; translated by Andreas Nestke
出版情報: Providence, R.I. : American Mathematical Society, c2000  xvi, 195 p. ; 27 cm
シリーズ名: Graduate studies in mathematics ; v. 25
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目次情報: 続きを見る
Clifford algebras and spin representation
Spin structures Dirac operators
Analytical properties of Dirac operators
Eigenvalue estimates for the Dirac operator and twistor spinors
Seiberg-Witten invariants
Principal bundles and connections
Bibliography
Index
Clifford algebras and spin representation
Spin structures Dirac operators
Analytical properties of Dirac operators
7.

図書
図書
7. Riemannian geometry and geometric analysis. 2nd ed
Jürgen Jost
出版情報: Berlin ; New York : Springer, c1998  xiii, 455 p. ; 24 cm
シリーズ名: Universitext
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8.

図書
図書
8. Riemannian geometry : a modern introduction. 1st pbk. ed. with corr (: pbk)
Isaac Chavel
出版情報: Cambridge [England] : Cambridge University Press, 1995  xii, 386 p. ; 23 cm
シリーズ名: Cambridge tracts in mathematics ; 108
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目次情報: 続きを見る
Preface to the Second Edition
Preface
Riemannian Manifolds / I：
Connections / I.1：
Parallel Translation of Vector Fields / I.2：
Geodesics and the Exponential Map / I.3：
The Torsion and Curvature Tensors / I.4：
Riemannian Metrics / I.5：
The Metric Space Structure / I.6：
Geodesics and Completeness / I.7：
Calculations with Moving Frames / I.8：
Notes and Exercises / I.9：
Riemannian Curvature / II：
The Riemann Sectional Curvature / II.1：
Riemannian Submanifolds / II.2：
Spaces of Constant Sectional Curvature / II.3：
First and Second Variations of Arc Length / II.4：
Jacobi's Equation and Criteria / II.5：
Elementary Comparison Theorems / II.6：
Jacobi Fields and the Exponential Map / II.7：
Riemann Normal Coordinates / II.8：
Riemannian Volume / II.9：
Geodesic Spherical Coordinates / III.1：
The Conjugate and Cut Loci / III.2：
Riemannian Measure / III.3：
Volume Comparison Theorems / III.4：
The Area of Spheres / III.5：
Fermi Coordinates / III.6：
Integration of Differential Forms / III.7：
Appendix: Eigenvalue Comparison Theorems / III.8：
Riemannian Coverings / IV：
The Fundamental Group / IV.1：
Volume Growth of Riemannian Coverings / IV.3：
Discretization of Riemannian Manifolds / IV.4：
The Free Homotopy Classes / IV.5：
Surfaces / IV.6：
Systolic Inequalities / V.1：
Gauss-Bonnet Theory of Surfaces / V.2：
The Collar Theorem / V.3：
The Isoperimetric Problem: Introduction / V.4：
Surfaces with Curvature Bounded from Above / V.5：
The Isoperimetric Problem on the Paraboloid of Revolution / V.6：
Isoperimetric Inequalities (Constant Curvature) / V.7：
The Brunn-Minkowski Theorem / VI.1：
Solvability of a Neumann Problem in R[superscript n] / VI.2：
Fermi Coordinates in Constant Sectional Curvature Spaces / VI.3：
Spherical Symmetrization and Isoperimetric Inequalities / VI.4：
M. Gromov's Uniqueness Proof - Euclidean and Hyperbolic Space / VI.5：
The Isoperimetric Inequality on Spheres / VI.6：
The Kinematic Density / VI.7：
The Differential Geometry of Analytical Dynamics / VII.1：
The Berger-Kazdan Inequalities / VII.2：
On Manifolds with No Conjugate Points / VII.3：
Santalo's Formula / VII.4：
Isoperimetric Inequalities (Variable Curvature) / VII.5：
Croke's Isoperimetric Inequality / VIII.1：
Buser's Isoperimetric Inequality / VIII.2：
Isoperimetric Constants / VIII.3：
Discretizations and Isoperimetry / VIII.4：
Comparison and Finiteness Theorems / VIII.5：
Preliminaries / IX.1：
H. E. Rauch's Comparison Theorem / IX.2：
Comparison Theorems with Initial Submanifolds / IX.3：
Refinements of the Rauch Theorem / IX.4：
Triangle Comparison Theorems / IX.5：
Convexity / IX.6：
Center of Mass / IX.7：
Cheeger's Finiteness Theorem / IX.8：
Hints and Sketches for Exercises / IX.9：
Hints and Sketches: Chapter I
Hints and Sketches: Chapter II
Hints and Sketches: Chapter III
Hints and Sketches: Chapter IV
Hints and Sketches: Chapter V
Hints and Sketches: Chapter VI
Hints and Sketches: Chapter VII
Hints and Sketches: Chapter VIII
Hints and Sketches: Chapter IX
Bibliography
Author Index
Subject Index
Preface to the Second Edition
Preface
Riemannian Manifolds / I：
9.

図書
図書
9. The Riemann legacy : Riemannian ideas in mathematics and physics
by Krzysztof Maurin
出版情報: Dordrecht ; Boston : Kluwer Academic, c1997  xxii, 717 p. ; 25 cm
シリーズ名: Mathematics and its applications ; v. 417
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10.

図書
図書
10. Comparison geometry (: hard. ; : pbk.)
edited by Karsten Grove, Peter Petersen
出版情報: Cambridge : Cambridge University Press, 1997  x, 262 p. ; 25 cm
シリーズ名: Mathematical Sciences Research Institute publications ; 30
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目次情報: 続きを見る
Scalar curvature and geometrization conjectures for 3-manifolds / Michael T. Anderson1：
Injectivity radius estimates and sphere theorems / Uwe Abresch ; Wolfgang T. Meyer2：
Aspects of Ricci curvature / Tobias H. Colding3：
A genealogy of noncompact manifolds of nonnegative curvature: history and logic / R. E. Greene4：
Differential geometric aspects of Alexandrov spaces / Yukio Otsu5：
Convergence theorems in Riemannian geometry / Peter Petersen6：
The comparison geometry of Ricci curvature / Shunhui Zhu7：
Construction of manifolds of positive Ricci curvature with big volume and large Betti numbers / G. Perelman8：
Collapsing with no proper extremal subsets / 9：
Example of a complete Riemannian manifold of positive Ricci curvature with Euclidean volume growth and with nonunique asymptotic cone / 10：
Applications of quasigeodesics and gradient curves / Anton Petrunin11：
Scalar curvature and geometrization conjectures for 3-manifolds / Michael T. Anderson1：
Injectivity radius estimates and sphere theorems / Uwe Abresch ; Wolfgang T. Meyer2：
Aspects of Ricci curvature / Tobias H. Colding3：
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